Are you asking what the answer is or how to prove it? You'll need to use the fact that the arc tangent function is defined to take the (principal) values (−π/2, π/2). Otherwise there will be multiple answers.

Firstly, you have to know how to draw the graph of such funtions as (log, tang, cot, sin, cos...) then try to draw the inverse graph of these functions. You should also know about limit and its theorems. That's how I try to find a way to solve that kind of problems.
Now your question's solution:
[itex]\lim_{x\rightarrow+∞}arctgx =?[/itex]
When you draw the graph of arctgx and take the limit x goes to positive infinity, the function converges [itex]\frac{\pi}{2}[/itex]. Eventually, we conclude the answer is
[itex]\lim_{x\rightarrow+∞}arctgx = \frac{\pi}{2}[/itex]